Cevian line

In triangle geometry, a Cevian line is a line in a triangle joining a vertex of the triangle to a point on the opposite side. A Cevian set is a set of three lines lines, one for each vertex. A Cevian set is concurrent if the three lines meet in a single point.

Ceva's theorem

Let the triangle be ABC, with the Cevian lines being AX, BY and CZ. Ceva's theorem states that the Cevian set is concurrent if and only if

${\frac {AZ}{ZB}}\cdot {\frac {BX}{XC}}\cdot {\frac {CY}{YA}}=-1.\,$

Concurrent sets

Examples of concurrent Cevian sets include:

• The altitudes, meeting at the orthocentre
• The medians, meeting at the centroid
• The angle bisectors, meeting at the incentre

References

• H.S.M. Coxeter; S.L. Greitzer (1967). Geometry revisited. MAA. ISBN 0-88385-619-0.